We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the prooftheoretic strength of the regularity of Lebesgue measure for Gδ ...

Abstract In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace operator. The sufficient conditions for summability is obtained. For the orders of Riesz means, which greater than critical index N−1 2 we established the estimation for maximal operator of the Riesz means. Note that when order α of Riesz means is less than cr...

Matrix recovery is raised in many areas. In this paper, we build up a framework for almost everywhere matrix recovery which means to recover almost all the P ∈ M ⊂ F from Tr(AjP ), j = 1, . . . , N where Aj ∈ Vj ⊂ F. We mainly focus on the following question: how many measurements are needed to recover almost all the matrices in M? For the case where both M and Vj are algebraic varieties, we us...